Just a bit of background info. If f:S^2 --> R is a continuous mapping, then as a special case of the Borsuk-Ulam theorem (you don't need to know this), there exists c in S^2 such that f(c) = f(-c). S^2 just means 2 sphere, which is just what it sounds like - a sphere. So basically, I have a continuous function f(x) =...blah.... such that f(c) = f(-c) for some c in the domain. Suppose f measures the intensity of sunlight at each point on the earth's surface(so S^2 is the earth's surface), then it follows (from what I wrote above), that there exists a pair of points opposite each other on the earth's surface at which sunlight intensity is the same, but if it is daytime at one point, it's night time at the point opposite! Resolve the paradox.

The intensity of sunlight would be equal at at least one set of antipodal points (this satisfies f(c)=f(-c)). I think you're looking for points at the "edge" of the sphere where light is incoming at a 90degrees angle. A bit like a city in the evening (light is incoming at about 90 degrees) and another city whose time zone is +/- 12hours and in the morning time where the light is also incoming at 90degrees.

There is no paradox. The illusion of the paradox is created because the claim that "but if it is daytime at one point, it's night time at the point opposite!" is false (or, perhaps, you can say that it is violated on a set of measure zero ). There can be a point at the day-night boundary and an antipodal point at the night-day boundary which can be paired. With an appropriate choice of the co-ordinate system, those points can be c and -c respectively. For your particular example, there is an infinity of such pairs.